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Mirrors > Home > ILE Home > Th. List > divcanap4d | GIF version |
Description: A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) |
Ref | Expression |
---|---|
divcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divclapd.3 | ⊢ (𝜑 → 𝐵 # 0) |
Ref | Expression |
---|---|
divcanap4d | ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divclapd.3 | . 2 ⊢ (𝜑 → 𝐵 # 0) | |
4 | divcanap4 8082 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | |
5 | 1, 2, 3, 4 | syl3anc 1172 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1287 ∈ wcel 1436 class class class wbr 3814 (class class class)co 5594 ℂcc 7269 0cc0 7271 · cmul 7276 # cap 7976 / cdiv 8055 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-mulrcl 7365 ax-addcom 7366 ax-mulcom 7367 ax-addass 7368 ax-mulass 7369 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-1rid 7373 ax-0id 7374 ax-rnegex 7375 ax-precex 7376 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-apti 7381 ax-pre-ltadd 7382 ax-pre-mulgt0 7383 ax-pre-mulext 7384 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-id 4087 df-po 4090 df-iso 4091 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-iota 4937 df-fun 4974 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-reap 7970 df-ap 7977 df-div 8056 |
This theorem is referenced by: mvllmulapd 8216 ltmuldiv 8247 irrmul 9041 mulqmod0 9640 modqcyc 9669 q2txmodxeq0 9694 expaddzaplem 9849 facdiv 9995 permnn 10028 cjdivap 10184 resqrexlemcalc1 10288 sqrtdiv 10316 absdivap 10344 gcddiv 10802 divgcdcoprm0 10877 hashgcdlem 10997 |
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